Rational and Irrational Numbers Lesson : Definition, Difference & Examples

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Updated: Jan 9, 2025

Lesson

Lesson Overview

What Are Rational and Irrational Numbers?
Key Differences Between Rational and Irrational Numbers
How to Classify Numbers as Rational or Irrational
Graphical Representation of Rational and Irrational Numbers on Number Line
Examples of Rational and Irrational Numbers
Rational and Irrational Numbers Assessment

What Are Rational and Irrational Numbers?

Rational Numbers
A rational number is any number that can be written as a fraction. It has a numerator and a denominator. The denominator cannot be zero.

Examples:

3 (can be written as 3/1)
1/2 (fraction)
0.75 (can be written as 3/4)
Irrational Numbers
An irrational number cannot be written as a fraction. These numbers have non-repeating and non-ending decimal values.

Examples:

√2 = 1.414… (it does not stop or repeat)
π = 3.14159… (it goes on forever without repeating)

Fig: Rational and Irrational numbers

Take This Quiz :

Rational Numbers & Irrational Numbers Quiz

How do you know if a number is rational or irrational? Is 4/5 rational or irrational? Check out our online quiz and attempt some amazing questions to unleash the maths expert.

Questions: 10 | Attempts: 1328 | Last updated: Jan 09, 2025

Sample Question

The decimal expansion of an irrational number may be

Terminating

Recurring

Either terminating or non- terminating

Non-terminating and non-recurring

Key Differences Between Rational and Irrational Numbers

Aspect	Rational Numbers	Irrational Numbers
Definition	Numbers that can be expressed as fractions p/q, where p and q are integers, and q ≠ 0.	Numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimals.
Examples	1/2, -3, 0, 4.75, 7/3	√2, √3, π, e, √5
Decimal Form	Terminates or repeats (e.g., 0.5, 0.333...)	Non-terminating and non-repeating (e.g., 1.414..., 3.141...)
Representation	Can be written in the form of a fraction.	Cannot be written in the form of a fraction.
Number Line	Can be exactly located on the number line.	Can only be approximately located on the number line.
Nature	Includes whole numbers, integers, and fractions.	Includes square roots of non-perfect squares, π, e, etc.
Occurrence	Finite or predictable decimal patterns.	Infinite and unpredictable decimal patterns.

How to Classify Numbers as Rational or Irrational

Check if it is a fraction.
- If yes, it is rational.
  Example: 7/8.
Check the decimal form.
- If the decimal stops or repeats, it is rational.
  Example: 0.333… (rational).
Check if it is a square root.
- If the square root gives a whole number, it is rational.
  Example: √4 = 2 (rational).
- If the square root is not exact, it is irrational.
  Example: √5 (irrational).

Take This Quiz :

Find which is rational and irrational numbers?

A rational number is a number that can be written as a ratio that means it can be written as a fraction. Irrational numbers are named such because it cannot be written as a...

Questions: 10 | Attempts: 213 | Last updated: Dec 30, 2024

Sample Question

1.23

Graphical Representation of Rational and Irrational Numbers on Number Line

1. Graphing Rational Numbers

Rational numbers are numbers that can be written as fractions p/q, where p and q are integers, and q is not zero. This includes whole numbers, integers, and fractions.

Steps:

Mark whole numbers: Draw a number line and label evenly spaced points like -3, -2, -1, 0, 1, 2, and so on.
Locate fractions: For a fraction like 1/2:
- Divide the section between 0 and 1 into two equal parts.
- Mark 1/2 at the midpoint.
Locate negative fractions: For -3/4:
- Divide the section between -1 and 0 into four equal parts.
- Mark the third part from 0 toward -1.

Example:

To plot 2/3, divide the section between 0 and 1 into three equal parts and mark the second part.
To plot -5/2, divide the section between -2 and -3 into two equal parts and mark the middle.

Fig: Graphing Rational Numbers on Number Line

2. Graphing Irrational Numbers

Irrational numbers are numbers that cannot be expressed as fractions. Their decimal forms are non-repeating and non-terminating, such as √2, √3, and π.

Steps:

Approximate the value: Find the approximate decimal value of the irrational number.
- √2 ≈ 1.41
- √3 ≈ 1.73
Mark the position: Locate the approximate value between two integers. For √2, mark slightly past 1 but before 1.5.
For negative values: Use the same method on the negative side. For -√3, mark slightly past -1.5 but before -2.

Example:

√2 lies between 1 and 2, closer to 1.4.
-√3 lies between -2 and -1, closer to -1.7.

Fig: Graphing Irrational Numbers on Number Line

3. Tips for Graphing

Use evenly spaced sections for accuracy.
Irrational numbers can only be approximated, so explain their exact values separately.
Fractions should divide sections on the line based on their denominators.

Examples of Rational and Irrational Numbers

Rational Numbers:
- Example 1: 3 is rational because it can be written as 3/1.
- Example 2: 0.75 is rational because it is 3/4.
Irrational Numbers:
- Example 1: √3 is irrational because it equals 1.732… (non-repeating).
- Example 2: π is irrational because it never ends and never repeats.

Rational and Irrational Numbers Assessment

Identify whether the following numbers are rational or irrational:
a) 4/5
b) √7
c) 2.5
d) 0.1010010001…
Place the following numbers on the number line:
a) 1/2
b) √2
c) -3